Posted in Algebra

Tough Algebra Questions about Equations and Expressions

Here are some questions your students have been wanting to ask you in your algebra class. Daniel Chazan and Michal Yerushalmy in their article On Appreciating the Cognitive Complexity of School Algebra posed these questions about equivalence of equations , solving equations, and equivalence of expressions for us teachers to ponder upon.

function_notationHow will you answer the following questions? What explanation will you give to the students?

Continue reading “Tough Algebra Questions about Equations and Expressions”

Posted in Algebra, Number Sense

The many faces of multiplication

The following table is not meant to be a complete list of ideas about the concept of multiplication. It is not meant to be definitive but it does include the basic concepts about multiplication for middle school learners. The inclusion of the last two columns about the definition of a prime number and whether or not 1 is considered a prime show that there are definitions adapted to teach school mathematics that teachers in the higher year levels need to revise. Note that branching and grouping which make 1 not a prime number can only model multiplication of whole numbers unlike the rest of the models. Multiplication as repeated addition has launched a math war. Formal mathematics, of course, has a definitive answer on whether 1 is prime or not. According to the Fundamental Theorem of Arithmetic, 1 must not be prime so that each number greater than 1 has a unique prime factorisation.

If multiplication is … … then a product is: … a factor is: … a prime is: Is 1 prime?
REPEATED ADDITION a sum (e.g., 2×3=2+2+2 = 3+3) either an addend or the count of addends a product that is either a sum of 1’s or itself. NO: 1 cannot be produced by repeatedly adding any whole number to itself.
GROUPING a set of sets (e.g., 2×3 means either 2 sets of three items or 3 sets of 2) either the number of items in a set, or the number of sets a product that can only be made when one of the factor is 1 YES: 1 is one set of one.
BRANCHING the number of end tips on a ‘tree’ produced by a sequence of branchings (think of fractals) a branching (i.e., to multiply by n, each tip is branched n times) a tree that can only be produced directly (i.e., not as a combination of branchings) NO: 1 is a starting place/point … a pre-product as it were.
FOLDING number of discrete regions produced by a series of folds (e.g., 2×3 means do a 2-fold, then a 3-fold, giving 6 regions) a fold (i.e., to multiply by n, the object is folded in n equal-sized regions using n-1 creases) a number of regions that can only be folded directly NO: no folds are involved in generating 1 region
ARRAY-MAKING cells in an m by n array a dimension a product that can only be constructed with a unit dimension. YES: an array with one cell must have a unit dimension

The table is from the study of Brent Davis and Moshe Renert in their article Mathematics-for-Teaching as Shared Dynamic Participation published in For the Learning of Mathematics. Vol. 29, No. 3. The table was constructed by a group of teachers who were doing a concept analysis about multiplication. Concept analysis involves tracing the origins and applications of a concept, looking at the different ways in which it appears both within and outside mathematics, and examining the various representations and definitions used to describe it and their consequences, (Usiskin et. al, 2003, p.1)

The Multiplication Models (Natural Math: Multiplication) also provides good visual for explaining multiplication.

You may also want to read How should students understand the subtraction operation?

Posted in Elementary School Math, Number Sense

Bob is learning calculation

Bob is an elementary school student. He is learning to calculate. He just learned about addition and multiplication but there are some things that he doesn’t understand. For example, how come 1+3 = 3 + 1? How can it be the same thought Bob? Every morning I have 1 piece of bread for breakfast while Dad has 3 pieces. If I have 3 pieces while Dad has 1 piece, I will be too full and Dad will be hungry?

When they added three numbers, Bob did not understand (1+2) + 1 = 1 + (2+1). Usually I like to drink 1 cup of coffee with 2 spoons of milk then afterwards have a piece of bread. I would not feel well if I first drink a cup of coffee then afterwards drink 2 spoons of milk while having 1 piece of bread. How come they are the same, thought Bob.

The most confusing part was after the lesson on fraction. Bob learned that 1/2 = 2/4. So when he got back home he tried to share 6 apples with his sister Linda. He divided the 6 apples into two groups – 2 apples in one group and 4 apples in another group.

apples, dividing apples

From the set of two apples he gave 1 to Linda. That’s 1/2. From the set of four apples, he took 2, that’s 2/4. It is equal he said. But Linda did not agree with him because she got 1 apple less that he. Bob thought, how can this be? Why would 1/2 = 2/4 not work for apples!

The next day, the teacher asked Bob to add 1/2 and 2/4? Bob wrote 1/2 + 2/4 = 3/6 because taking 1 apple from 2 apples then 2 apples from 4 apples, he must have taken a total of 3 apples from 6 apples!

This story is adapted from A Framework of Mathematical Knowledge for Teaching by J. Li, X. Fan, and Y Zhui at the EARCOME5 2010 conference.

Point for reflection:

What has Bob missed about the meaning of addition of natural numbers? the meaning of fraction?

You may want to read the following posts about math knowledge for teaching: